although homotopy-based methods, namely homotopy analysis method andhomotopy perturbation method, have largely been used to solve functionalequations, there are still serious questions on the convergence issue of thesemethods. some authors have tried to prove convergence of these methods, butthe researchers in this article indicate that some of those discussions are faulty.here, after criticizi...

let  be a banach algebra. let  be linear mappings on . first we demonstrate a theorem concerning the continuity of double derivations; especially that all of -double derivations are continuous on semi-simple banach algebras, in certain case. afterwards we define a new vocabulary called “-higher double derivation” and present a relation between this subject and derivations and finally give some ...

The aim of this paper is to prove the Uniform Boundedness Principle and Banach-Steinhaus Theorem for anti linear operators and hence strong linear operators on Banach hypervector spaces. Also we prove the continuity of the product operation in such spaces.

1. Basic definitions 2. Riesz’ Lemma 3. Counter-examples for unique norm-minimizing element 4. Normed spaces of continuous linear maps 5. Dual spaces of normed spaces 6. Baire’s theorem 7. Banach-Steinhaus/uniform-boundedness theorem 8. Open mapping theorem 9. Closed graph theorem 10. Hahn-Banach theorem Many natural spaces of functions, such as C(K) for K compact, and C[a, b], have natural str...

in this paper, a vector version of the intermediate value theorem is established. the main theorem of this article can be considered as an improvement of the main results have been appeared in [textit{on fixed point theorems for monotone increasing vector valued mappings via scalarizing}, positivity, 19 (2) (2015) 333-340] with containing the uniqueness, convergent of each iteration to the fixe...

As a cornerstone of functional analysis, Hahn–Banach theorem constitutes an indispensable tool of modern analysis where its impact extends beyond the frontiers of linear functional analysis into several other domains of mathematics, including complex analysis, partial differential equations and ergodic theory besides many more. The paper is an attempt to draw attention to certain applications o...

We show that if X is a separable Banach space (or more generally a Banach with an infinite-dimensional separable quotient) then there is a continuous mapping f : X → X such that the autonomous differential equation x′ = f(x) has no solution at any point. In order to put our results into context, let us start by formulating the classical theorem of Peano. Theorem 1. (Peano) Let X = R, f : R×X → ...

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael’s Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse o...

1. The norm || • || in a Banach algebra A is said to be minimal [l ] if, given any other norm || •||1 in A (with respect to which A need not be complete), the condition ||a||iá||a|| for each oG^4 implies that ||a[|i = ||a||. We shall say that || •|| is absolutely minimal if, given any other norm ||-||i whatever in A, then ||a||iè||a|| for each aEA. An absolutely minimal norm is of course minima...

Let C be a nonempty closed convex subset of a smooth Banach space E and let A be an accretive operator of C into E. We first introduce the problem of finding a point u ∈ C such that 〈Au, J(v− u)〉 ≥ 0 for all v ∈ C, where J is the duality mapping of E. Next we study a weak convergence theorem for accretive operators in Banach spaces. This theorem extends the result by Gol’shteı̆n and Tret’yakov i...